The generalized recurrent set, explosions and Lyapunov functions

Olga Bernardi (1) , Anna Florio (2) , Jim Wiseman (3)

1 Dipartimento di Matematica Pura e Applicata [Padova]
2 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
3 Department of Mathematics, Agnes Scott College

We consider explosions in the generalized recurrent set for homeomorphisms on a compact metric space. We provide multiple examples to show that such explosions can occur, in contrast to the case for the chain recurrent set. We give sufficient conditions to avoid explosions and discuss their necessity. Moreover, we explain the relations between explosions and cycles for the generalized recurrent set. In particular, for a compact topological manifold with dimension greater or equal 2, we characterize explosion phenomena in terms of existence of cycles. We apply our results to give sufficient conditions for stability, under C 0 perturbations, of the property of admitting a continuous Lyapunov function which is not a first integral.

Domains: Mathematics [math]; Dynamical Systems [math.DS]

Olga Bernardi, Anna Florio, Jim Wiseman. The generalized recurrent set, explosions and Lyapunov functions. Journal of Dynamics and Differential Equations, 2020. ⟨hal-02097706⟩